Langlands Functoriality and $L$-Homomorphisms

An $L$-homomorphism is a group homomorphism $\omega:{}^LG'\to {}^LG$ such that:

• it commutes with the projections to the Galois/Weil groups, and
• its restriction to dual groups is an algebraic homomorphism $\widehat G'\to \widehat G$.

Functoriality (as used in the letter): If $\phi'$ has Satake parameters $\alpha'_p$ for $G'$, then the predicted transfer $\phi$ for $G$ should satisfy (for almost all $p$)

as conjugacy classes in ${}^LG$.